On profinite groups in which commutators are covered by finitely many subgroups

TL;DR

This paper investigates profinite groups where certain word-values, especially commutators, are covered by finitely many subgroups, establishing properties of the verbal subgroup based on the properties of these covering subgroups.

Contribution

It proves that if all $w$-values in a profinite group are contained in finitely many subgroups with specific properties, then the corresponding verbal subgroup inherits those properties.

Findings

Verbal subgroup $w(G)$ shares properties of covering subgroups.

Finite exponent of subgroups implies bounded exponent of $oldsymbol{ ext{γ}_k(G)}$.

Finite rank of subgroups implies bounded rank of $oldsymbol{ ext{γ}_k(G)}$.

Abstract

For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $\gamma_k$-values in $G$, it is shown that $\gamma_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contains all $\gamma_k$-values, it is shown that $\gamma_k(G)$ has finite $(k,r,s)$-bounded rank.